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We introduce the problem of maximizing approximately k-submodular functions subject to size constraints. In this problem, one seeks to select k-disjoint subsets of a ground set with bounded total size or individual sizes, and maximum utility, given by a function that is "close" to being k-submodular. The problem finds applications in tasks such as sensor placement, where one wishes to install k types of sensors whose measurements are noisy, and influence maximization, where one seeks to advertise k topics to users of a social network whose level of influence is uncertain. To deal with the problem, we first provide two natural definitions for approximately k-submodular functions and establish a hierarchical relationship between them. Next, we show that simple greedy algorithms offer approximation guarantees for different types of size constraints. Last, we demonstrate experimentally that the greedy algorithms are effective in sensor placement and influence maximization problems.

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High altitude parabolic is difficult to identify because of its small size, fast speed and changeable state, which makes it difficult for subsequent forensics and accountability. This paper proposes a high-altitude parabolic detection and tracking method GPK-YOLOv5s, which integrates Content-Aware Reassembly of Features (CARAFE) and self-attention to realize parabolic detection and tracking. For the detection network, the backbone integrates C3Ghost module to extract effective features and simplify the network. C3Transformer module is embedded in the feature extraction and fusion layer to pay attention to the global context information. The feature fusion layer uses CARAFE module for up sampling to perceive effective features, and integrates shallow features and deep features to form a new large-scale detection layer (Output4) to further obtain smaller receptive fields. Improved multi-scale detection heads are embedded with CBAM to enhance the expression ability of targets. To overcome the frame loss of real-time detection, improved multiscale detection heads are externally connected with Kalman filter to track targets. This experiment verifies that the detection Precision, Recall and F1 value of GPK-YOLOv5s reached 99.0%, 98.6% and 98.8% respectively, which are 2.8%, 4.1% and 3.5% higher than YOLOv5s respectively. And GPK-YOLOv5s is lighter, and the calculation consumption is reduced by 0.4 GFLOPs.

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